Ebook: High Dimensional Probability II

High dimensional probability, in the sense that encompasses the topics rep­ resented in this volume, began about thirty years ago with research in two related areas: limit theorems for sums of independent Banach space valued random vectors and general Gaussian processes. An important feature in these past research studies has been the fact that they highlighted the es­ sential probabilistic nature of the problems considered. In part, this was because, by working on a general Banach space, one had to discard the extra, and often extraneous, structure imposed by random variables taking values in a Euclidean space, or by processes being indexed by sets in R or Rd. Doing this led to striking advances, particularly in Gaussian process theory. It also led to the creation or introduction of powerful new tools, such as randomization, decoupling, moment and exponential inequalities, chaining, isoperimetry and concentration of measure, which apply to areas well beyond those for which they were created. The general theory of em­ pirical processes, with its vast applications in statistics, the study of local times of Markov processes, certain problems in harmonic analysis, and the general theory of stochastic processes are just several of the broad areas in which Gaussian process techniques and techniques from probability in Banach spaces have made a substantial impact. Parallel to this work on probability in Banach spaces, classical proba­ bility and empirical process theory were enriched by the development of powerful results in strong approximations.

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Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Moment Bounds for Self-Normalized Martingales....Pages 3-11
Exponential and Moment Inequalities for U-Statistics....Pages 13-38
A Multiplicative Inequality for Concentration Functions of n-Fold Convolutions....Pages 39-47
On Exact Maximal Khinchine Inequalities....Pages 49-63
Strong Exponential Integrability of Martingales with Increments Bounded by a Sequence of Numbers....Pages 65-76
Front Matter....Pages 77-77
On Uniform Laws of Large Numbers for Smoothed Empirical Measures....Pages 79-87
Weak Convergence of Smoothed Empirical Processes: Beyond Donsker Classes....Pages 89-105
Limit Theorems for Smoothed Empirical Processes....Pages 107-113
Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes....Pages 115-133
Front Matter....Pages 135-135
A Note on the Gaussian Correlation Conjecture....Pages 137-161
Probability Estimates for Lower Levels of Certain Gaussian Processes with Stationary Increments....Pages 163-171
Front Matter....Pages 173-179
Asymptotic Independence of the Local Empirical Process Indexed by Functions....Pages 181-181
The Az?ma-Yor Embedding in Brownian Motion with Drift....Pages 183-205
A New Way to Obtain Estimates in the Invariance Principle....Pages 207-221
Front Matter....Pages 223-245
On the Law of the Iterated Logarithm for Local Times of Recurrent Random Walks....Pages 247-247
A General Compact Law of the Iterated Logarithm in Banach Spaces....Pages 249-259
Front Matter....Pages 261-278
Dominating Points for General Sequences of Random Vectors....Pages 279-279
Large Deviations for Local Empirical Measures....Pages 281-291
Front Matter....Pages 293-312
An Example Concerning Tightness of Sums of B-Valued Random Variables....Pages 313-313
Front Matter....Pages 315-327
Images and Level Sets of Additive Random Walks....Pages 313-313
Front Matter....Pages 329-345
Lee-Yang Models, Selfdecomposability and Negative-Definite Functions....Pages 347-347
When an Isotropic Random Vector is Pseudo-Isotropic....Pages 349-357
Support Fragmentation for Multiplicative Cascade Measures....Pages 359-366
On Simulating Fractional Brownian Motion....Pages 367-376
Front Matter....Pages 377-387
On Robust Recursive Nonparametric Curve Estimation....Pages 389-389
Variable Kernel Estimates: on the Impossibility of Tuning the Parameters....Pages 391-403
Almost Sure Asymptotic Optimality of Cross Validation for Spline Smoothing....Pages 405-424
Rademacher Processes and Bounding the Risk of Function Learning....Pages 425-441
Front Matter....Pages 443-457
Bootstrapping Empirical Distributions under Auxiliary Information....Pages 459-459
On the Characteristic Function of the Matrix von Mises—Fisher Distribution with Application to SO(N)—Deconvolution....Pages 461-476
Testing for Ellipsoidal Symmetry of a Multivariate Distribution....Pages 477-492
Back Matter....Pages 493-510
....Pages 511-512